Is there a way to calculate the advanced solution from a fundamental solution. Namely for $P$ a partial differential operator and $E$ a fundamental solution with $\text{supp} \ E =\mathbb{R} \times \mathbb{R}^n $ $$ P(E) = \delta \ .$$ can I find a solution $E_+$ with $\text{supp} \ E_+ = \mathbb{R}^+ \times \mathbb{R}^n$ such that $P(E_+) = \delta$ ?
I know that in the reversed case $\tilde{E} := \frac{E_++E_-}{2}$ fullfills $P(\tilde{E}) = \delta$ .
For $H_t(t,x) := 1$ if $t \geq 0$ and $H_t(t,x) := 0$ if $t < 0$ I thought that maybe $E \ H_t$ could yield $$ \left<P(E \ H_t ),\phi\right> = \left<P(E) H_t + E \ P(H_t),\phi\right> \\ = \left< \delta, \phi\right> \left< H_t, \phi\right> + \left<E,\phi\right> \left<P(H_t), \phi\right> . $$ But then I don't know how to continue and whether this works for all fundamental or just the one gained from the fourier transform or not at all. Does anybody have a hint?